Horseshoe dynamics

In this section, we will introduce the Melnikov integral, which estimates the gaps between stable and unstable manifolds. Thus, if the Melnikov integral attains the value zero, this signals the existence of intersections between stable and unstable manifolds. If the intersection is transversal, it implies the existence of horseshoe dynamics [32], that is, chaotic behavior. On the other hand, if the intersection is tangent [11], it implies that the system is at a transition between different kinds of dynamics [12]. Such transitions of chaotic behavior are called crisis [13]. The tangency will be further analyzed in Section VII. 

Equation (64) shows that the distance d x,a) exhibits an oscillatory dependence as a function of x. In other words, d x, a) changes between plus and minus values as initial conditions shift on the separatrix. This means that the stable and unstable manifolds have transverse intersections. See Fig. 14 showing how the oscillatory change of the integral implies the occurrence of transverse intersections. The existence of transverse intersections between stable and unstable manifolds leads to horseshoe dynamics—that is, chaos. Thus, the Melnikov integral given by Eq. (64) indicates that this system exhibits chaotic behavior. 

A generalized oscillator-wave model is considered showing that the inhomogeneous external influence is realized naturally and does not require any specific conditions. The article considers also the presence of a small horseshoe in the dynamics of a particle under the action of two waves. Originally the problem comes from the plasma physics in despite of the existence of some other applications of the differential equation we study here. 

Presence of a small horseshoe in the dynamics of a particle under the action of two waves... 

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

Also interesting is the dynamical behavior associated with the fixed point at infinity, that is, q,p) = (oo,0). Here we introduce the concept of homoclinic orbit, which is a trajectory that goes to an unstable fixed point in the past as well as in the future. A homoclinic orbit thus passes the intersection between the unstable and stable manifolds of a particular fixed point. Indeed, as shown in Fig. 6, these manifolds generate a so-called homoclinic web. In particular. Fig. 6a displays a Smale horseshoe giving a two-symbol subdynamics, indicating that the fixed point (oo,0) is not a saddle. Nevertheless, it is stUl unstable with distinct stable and unstable manifolds, with its dynamics much slower than that for a saddle. Figure 6b shows an example of a numerical plot of the stable and unstable manifolds. 

Jan Vrbik (1998). Novel analysis of tadpole and horseshoe orbits. Celestial Mechanics and Dynamical Astronomy 69, 283-291. 

Recent developments of plastic stents have aimed to improve the resistance of plastic stents to external compression forces. Therefore, metal has been incorporated into the plastic material of the stent. One of the latest developments is the dynamic bifurcation stent made of silicone (Freitag et al. 1994). This Dynamic stent (Riisch, Kernen, Germany) is reinforced with horseshoe-shaped steel struts. A posteriorly located flexible membrane allows dynamic compression of the stent during coughing, whereas the steel struts prevent airway compression from external forces. Theoretically, this stent mimics the mechanical dynamics of the normal trachea. The distal end is a Y shape which rides on the carina to prevent distal migration. 

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... 

Homburg, A. J., Kokubu, H. and Krupa, M. [1994] The cusp horseshoe and its bifurcations in the unfolding of an inclination flip homoclinic orbit, Ergod, TL Dynam. Syst 14, 667- 93. 

---